assessing algebra in the senior phase: a practical guide

Assessing Algebra in theSenior Phase:

A Practical Guide

A S S E S S M E N T I N P R A C T I C E

Assessing Algebra in the Senior Phase:

A Practical Guide

This booklet demonstrates how assessment can be used

in learning and teaching Mathematics in

the Senior Phase.

Acknowledgments

AuthorsCatherine Hunter; Sue Jobson

Original reference groupSiza Shongwe; Erna Lampen; David Adler; Jenny Glennie; Sue Cohen

Critical readersJackie Scheiber; Bronwen Wilson-Thompson

ArtworkDorianne van Noort

Design and layoutDesign Intervention

ISBN No: 978-0-9869837-3-3

This work is licensed under the Creative Commons Attribution 3.0 Unported licence.To view a copy of this licence visit http://creativecommons.org/licenses/by/3.0/

©Saide2010

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Contents

1. Introduction

2. The Mathematics Focus

3. The Assessment Focus

3.1 The purpose of assessment 8

3.2 The teaching and learning cycle 8

4. Let’s get Practical

4.1 The teaching and learning cycle in Mrs Mothae’s class 11

4.2 Assessment in Mrs Mothae’s Maths class 12

� Analysing Assessment Task 1 12

� Marking Assessment Task 1 18

� Recording and interpreting results 26

� Using the assessment results to plan 31

� The cycle is on-going 32

5. Appendices

Appendix 1a: Assessment Task 1 34

Appendix 1b: Answers for Assessment Task 1 36








About Assessing Algebra in the SeniorPhase: A Practical Guide

This booklet was first developed in 2004, shortly after the introduction of the Revised

National Curriculum Statement in South Africa. The intention of the booklet was to

help teachers of senior phase (junior secondary) to integrate assessment into their

teaching and learning.

Although the curriculum in South Africa has now changed, the purpose of assessment has not,

and we think that teachers will find very useful the practical approach to planning,

implementing and reflecting on assessment results.

As the booklet is an Open Educational Resource, we invite teachers or teacher educators to

take, use and adapt it for whatever curriculum they are currently teaching. All we request is

that you acknowledge the document as originally developed by Saide.

We would also like to hear what you think of this booklet, and especially how you use it.

Please contact us at [email protected].


A S S E S S I N G A L G E B R A I N T H E S E N I O R P H A S E : A P R A C T I C A L G U I D E

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1. Introduction

In this booklet, we look at assessment in the Mathematics classroom. We focus

particularly on algebra in the Senior Phase. Our main aim is to place assessment

where it belongs – as an integral part of the whole teaching and learning cycle. We

use some of the work of a teacher, Mrs Mothae, to illustrate some ideas about assessment

and to help you to reflect on your own assessment practice.

We aim to provide you with resources to enable you to:

� Set tasks to assess whether a learner has achieved intended outcomes

� Assess learners’ work, and record and interpret their results

� Use the results of assessment tasks to inform the ongoing teaching and learning process

Look out for these icons:

Task:Complete an activity

Comment:An idea is explained or described.

Reflect:Discuss and record your thoughts and ideas


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2. The Mathematics Focus of this Booklet

In this booklet we focus on an outcome that is common to many Maths curricula:

This was Learning Outcome 2 in the South African National Curriculum Statement, and we think it is ameaningful way to introduce algebra in the Senior Phase.

The learner is able to recognise, describe and represent patterns andrelationships, and solve problems using algebraic language and skills.

Have a look at Table 1 on the next page. The outcome is unpacked into so-called'assessment standards' for each of the grades in the Senior Phase.

� Do you notice how similar the Assessment Standards are across the grades?� Do you notice how the Assessment Standards in each grade build onto the

previous grade’s work?� In general terms, what would you need to teach your learners in order for

them to achieve these Assessment Standards? What do the AssessmentStandards expect learners to be able to do as they progress toward theoutcome?

But what havepatterns got to

do with algebra?

I thought algebra was solving equations

with letters like x and y

Well, patterns provide a meaningful reason to use letter symbols. When learners completea pattern, and then describe it, this enables them to find a general rule for the pattern.Then they can use letter symbols to put that general rule into mathematical language.

In the South African curriculum, learners are introduced to the teaching of algebra throughpatterns. Although there are other ways of introducing learners to algebra, this way hasbeen found to be very successful.



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Learning Outcome 2: The learner is able to recognise, describe and represent patterns andrelationships, and solve problems using algebraic language and skills

Grade 7 Grade 8 Grade 9We know this when the learner: We know this when the learner: We know this when the learner:

� Investigates and extends numeric � Investigates and extends numericand geometric patterns looking and geometric patterns lookingfor a relationship or rules, for a relationship or rules,including patterns: including patterns:

� Represented in physical or � Represented in physical ordiagrammatic form diagrammatic form

� Not limited to sequences � Not limited to sequencesinvolving constant difference involving constant differenceor ratio or ratio

� Found in natural and � Found in natural and cultural contexts cultural contexts

� Of the learner’s own creation � Of the learner’s own creation

� Represented in tables � Represented in tables� Represented algebraically

� Describes, explains and justifies � Describes, explains and justifiesobserved relationships or rules observed relationships or rulesin own words in own words or algebraically

� Represents and uses relationships � Represents and uses relationshipsbetween variables in order to between variables in order todetermine input and/or output determine input and/or output values in a variety of ways using: values in a variety of ways using:� verbal descriptions � verbal descriptions� flow diagrams � flow diagrams� tables � tables

� formulae, equations and � Constructs mathematical models expressions

that represent, describe and provide solutions to problem � Constructs mathematical models situations, showing responsibility that represent, describe and toward the environment and the provide solutions to problem health of others (including situations, showing responsibility problems within human rights, toward the environment and the social, economic, cultural and health of others (includingenvironmental contexts). problems within human rights,

social, economic, cultural and environmental contexts).

� Investigates in differentways, a variety of geometricand numeric patterns andrelationships by representingand generalising them andby explaining and justifyingthe rules that generate them(including patterns found innatural and cultural formsand patterns of the learner’sown creation)

� Represents and usesrelationships betweenvariables in order todetermine input and/oroutput values in a variety ofways using:� verbal descriptions

� flow diagrams

� tables

� formulae, equations and

expressions

� Constructs mathematicalmodels that represent,describe and providesolutions to problemsituations, showingresponsibility toward theenvironment and the healthof others (includingproblems within humanrights, social, economic,cultural and environmentalcontexts).

Table 1: Assessment standards for an introduction to algebraNote: This table gives learning outcome 2 with the relevant assessment standards from the South African National Curriculum

Statement. It can, however, be adapted for other national curricula.


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When we looked at the Assessment Standards in Table 1, we identified severalteaching and learning steps that are needed if learners are to achieve the standards:

1. Learners need to be able to complete patterns, by finding relationships betweennumbers. They also need to be able to complete patterns that are represented intables.

2. Then they need to be able to describe these patterns verbally – this leads to‘generalising’ or looking for the ‘general rule’.

3. Then they need to be able to use letter symbols to write down their general rules ina mathematical way. They also sometimes need to use flow diagrams and graphs todo this.

4. Lastly, they need to use these steps to solve a problem.

The example below illustrates a problem that takes learners from patterns, to making rules,to using variables (expressed as letter symbols) to solve a problem:

Sharing out the bread

Mrs Moroko bakes a loaf of bread.

Drawing provided

If she has to divide it between only herself and Mr Moroko, she would make 1cut in the bread:

However, if her daughter comes to visit her that day, she would need to make2 cuts in the bread in order to share it between 3 of them:

Her daughter brings her granddaughter with her! Now she needs to make 3cuts in the bread to share it between 4 of them.



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Complete this table:

� Write down a sentence to explain what the pattern or general rule is for working outhow many cuts Mrs Moroko must make.

� If Mrs Moroko wants to share the bread between people, how many cuts must shemake?

� If Mrs Moroko makes cuts in the bread, to how many people can she offer a piece of

Number of people

Number of cuts

2

1

3

2

4

3

7

?

10

?

?

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In this problem, there are two things that change – the number of cuts and thenumber of people. These are called the variables. Learners should be able todiscover the general rule that the number of cuts is always one less than thenumber of people. They should be able to write this rule using letter symbolsfor the variables. This would be = -1

bread?


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3. The Assessment Focus of this Booklet

3.1 The Purpose of Assessment

Even though the purposes for assessment of learners are many and varied, the main purpose is to givelearners opportunities for growth and development. This includes assessing learners to facilitate theirlearning.

1. Plan

In planning our work, we start by asking: What do the learners need to know? In otherwords, what outcomes do we want them to achieve? We ask ourselves: What learningopportunities should we provide for learners so that they can achieve the learning, andproduce evidence to show that they have achieved the outcome?

Planning involves three steps:� Consult your term or year plan to see which learning outcomes/objectives to use for

teaching and assessing. You need to plan your teaching to cover concrete, clearlydefined assessment standards or sub-topics within the broad topic oroutcome/objective.

� Plan a series of lessons, with teacher input and learner activities, that will give learnersthe opportunity to develop the knowledge and skills described in the assessmentstandards or sub-topics.

� Design an assessment task that gives learners a chance to prove (give evidence) thatthey have learnt and achieved the outcomes or mastered the topics you selected.

2. Facilitate Learning

In this part of the cycle, learners have an opportunity to work toward the outcomes. Thelearners do the learning activities that you have chosen. This is an ongoing process thatincludes teaching (giving them input), mediating their learning, observing their progressand assisting them to achieve the outcomes.

This step will equip the learners with the knowledge, skills and values needed in order toachieve the outcomes.

What this means is that assessment is only effective and purposeful if it is an integral part of theentire teaching and learning cycle. This is what we focus on in this booklet.

3.2 The teaching and learning cycle

How do we make assessment an integral part of the teaching and learning process? The steps described below suggest how we can work with an ongoing cycle of planning, facilitatinglearning and assessing.



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3. Assess

When the learners have had enough opportunity to acquire the skills and knowledge thatyou have planned, they are ready to complete an assessment task. The task should enableyou to assess how far the learners are in achieving the outcomes.

In this part of the cycle you also:

� Record the results of the assessment.

� Analyse the group’s results to see how they can be helped further towards achievingthe outcomes and to inform your planning for the next step of learning needed. It isimportant to note trends in the class as a whole, their strengths and weaknesses.

� You provide feedback to learners to help them identify their own problem areas. If alearner can identify what her problem is, she can begin to work on improving herknowledge and skills. Feedback must be given in a spirit of encouragement andsupport, emphasising what the learner is able to do and building on this. Negativecriticism can just entrench negative attitudes in learners.

� At times, report this assessment to parents, staff or others who need to know.

4. Reflect and Decide

In this part of the cycle you use learners’ responses to reflect on your own teaching, andto plan the details of the next step of the Learning Programme. You may need to think ofalternative ways to make the learning easier, provide more learning tasks or redo tasks toassist those learners who have not yet achieved the outcomes.

� Do you think that the cycle described above reflects your understanding ofassessment?

� Are here any other steps you think should be included?

� In what ways do you think this cycle reflects what you currently do in yourclassroom?


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Th

e T

ea

chin

g a

nd

Le

arn

ing

Cy

cle 1. Plan

Choose learning outcomes/topics and the more detailedassessment standards or sub-topics. � What must learners know and be able to do?� What are the outcomes?� What learning opportunities can you provide?� How can you assess what learners have achieved?

3. Assess

The learners’ achievement is assessed and resultsrecorded and interpreted. � Record your assessment.� Have the learners achieved the intended outcomes? � Identify strengths and weaknesses.� Provide constructive feedback.

4. Let’s get Practical

2. Facilitate Learning

Teaching and learning takes place. � Learners engage in learning activities. � You assess learners informally while you teach,

observe and mediate.

4. Reflect and Decide

If the group of learners has satisfactory results, you canmove to the next step in your plan. Individuals or thewhole group may need concepts reinforced and anotheropportunity to be assessed in some way.



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Meet Ms Sibongile Mothae. She teaches Maths at Rainbow Park SecondarySchool. This is how she has used the teaching and learning cycle.

“First, I used my year plan to plan in more detail for the term. We had planned to focus onLearning Outcome 2: 'The learner is able to recognise, describe and represent patterns andrelationships, and solve problems using algebraic language and skills' for this term. Iagreed with the four key steps described on page 10 and used them in my planning.

I looked at how this outcome was unpacked in my curriculum (see Table 1) across Grades 7, 8,and 9. I thought about what I wanted my Grade 8 learners to know and identifiedAssessment Standards that they needed to achieve. After designing learning activities, I setfour tasks that would assess whether learners had achieved these Assessment Standards atcertain key points in the programme.”

“For two weeks we worked with number and shape patterns in class. Thelearners completed patterns in a range of contexts and designed their ownpatterns, too. At first, I needed to help many of them. I also found that theyneeded to be reminded about how to use decimal fractions. I made notes abouttheir progress as I went along.”

“When I was confident that learners were ready to consolidate their work, I gavethem Assessment Task 1 to complete. Then I marked their work, recorded theirresults and gave feedback. I considered the strengths and weaknesses ofindividuals, and suggested what they needed to do to improve their work.”

“I then looked at the ways in which the class as a whole had managed thequestions in the task. I thought about why many had problems with certainquestions, and what I could do to help. I changed my original plan toaccommodate these ideas.”

COMPLETEPATTERNS

DESCRIBE ANDGENERALISE

USE LETTERSYMBOLS

SOLVEPROBLEMS

Task 1: Task 2: Task 3: Task 4:

4.1 The teaching and learning cycle in Mrs Mothae’s class


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4.2 Assessment in Mrs Mothae’s Maths class

In the previous section we saw how Mrs Mothae used the teaching and learning cycle in her work. In therest of this booklet we will consider in particular on Steps 3 and 4 of this cycle. In this section we paygreatest attention to step 3 – assessment. Our discussion focuses on Mrs Mothae’s use of AssessmentTask 1.

� ANALYSING ASSESSMENT TASK 1

All assessment tasks need to be carefully planned to:� assess what was stated as the outcomes of the teaching and learning at the level that is

expected from your grade. � include enough questions that allow the “average” learner to succeed, with some

challenging questions to extend the learners. � provide opportunities for you to diagnose problem areas and consolidate previous

Maths knowledge.

Mrs Mothae tried to set tasks that would put all of these principles into practice! We haveincluded Task 1 and the other three Assessment Tasks in appendices in a size suitable foryou to copy for your own class if you wish. In this booklet, however, we will only considerhow Mrs Mothae uses Assessment Task 1 to assess her learners, to analyse their strengthsand weaknesses and to adjust her Learning Programme based on this information. For easyreference, we have shown Assessment Task 1 on the next page.

Do Assessment Task 1

1. A good way for you to get to grips with Assessment Task 1 is for you to do ityourself. So, as a first step, you should read through the task and answer thequestions yourself.

2. As you do the task, think about some of the problems that might face yourlearners when they do the task.


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Learners need to understand simple fractions to be able to do these questions. Questions 1(c), (d) and (e)They also need to multiply and divide accurately.

Learners need to understand decimals to be able to do these questions.

Most learners in the Intermediate Phase should be able to complete these simple addition and subtraction patterns with little difficulty.

To complete this pattern to the left, learners need to divide by 4.

To complete the pattern to the left, learners need to multiply by 3; to complete the pattern to the right, they need to divide by 3 at each step.

Learners must be able to add simple fractions. Even learners who are able to add quarters might not show the consistency in the pattern and fill in the zero (some may put one eighth as their answer)*.

Learners must be able to add three digit numbers. There is a constant difference of 111 between consecutive numbers in the pattern. A good learner would show an understanding of place value and fill in 900 as the first number in the pattern.

Some learners may have looked for the patterns in the digits instead of the whole numbers. The first digits are 7, 6, and 5; the second digits are 8,7 and 6 respectively. However, continuing the pattern in this way to the left would lead to the incorrect answer.

This question needs an understanding of place value in large numbers.

This question tests the learner’s ability to be consistent in the pattern, but this time by subtracting zero from one to find the answer to the left of one. The result is that there are two consecutive ‘1s’ in the pattern.

In this question, learners have to show they can continue patterns in two dimensions or directions simultaneously.

In these questions, the learner needs to explain the patterns, instead of just using the pattern. This is a necessary skill before being able to use variables in algebra.

Learners have to extend shape patterns and match them to number patterns

Think About Assessment Task 1

1. Which questions In Assessment Task 1 did you find surprising, interesting,challenging or easy?

2. What can Assessment Task 1 tell you about the learner’s ability to meet theAssessment Standards from Learning Outcome 2 (listed on page 5)?

3. What other Maths knowledge does the learner need in order to completeAssessment Task 1?

4. Suggest different ways in which you could use this assessment task in yourclassroom.

5. Match each of these statements about Task 1 correctly with the questions from Task 1. Note: There may be more than one answer for each statement and the same questionfrom the task may apply to more than one of the statements. We have matched the firststatement with the questions as an example.

6. Review your answers to questions 2 - 4 in the light of these statements.



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Here are some of our ideas about the questions asked on Assessment Task 1. Compare them with yours.

Learners need to understand simple fractions to be able to do these questions Questions 1(c), (d) and (e)They also need to multiply and divide accurately.

Learners need to understand decimals to be able to do these questions Question 1(i)

Learners continue simple addition and subtraction patterns. Most learners in Question 1(a) and (b)Intermediate Level should be able to complete these patterns with little difficulty.

To complete this pattern to the left, learners need to divide by 4 Question 1(c)

To complete the pattern to the left, learners need to multiply by 3; to complete Question 1(d)the pattern to the right, they need to divide by 3 at each step

Learners must be able to add simple fractions. Even learners who are able to add Question 1(e)quarters might not show the consistency in the pattern and fill in the zero (some may put one eighth as their answer).

Learners must be able to add three digit numbers. There is a constant difference Question 1(f)of 111 between consecutive numbers in the pattern. A good learner would show an understanding of place value and fill in 900 as the first number in the pattern.

Some learners may have looked for the patterns in the digits instead of the whole Question 1(f)numbers. The first digits are 7, 6, and 5; the second digits are 8,7 and 6 respectively. However, continuing the pattern in this way to the left would lead to the incorrect answer.

This question needs an understanding of place value in large numbers. Question 1(g)

This question tests the learner’s ability to be consistent in the pattern, but this time by subtracting zero from one to find the answer to the left of one. The result is that there are two consecutive ‘1s’ in the pattern. Question 1(h)

In this question, learners have to show they can continue patterns in two Question 2(a) dimensions or directions simultaneously.

In these questions, the learner needs to explain the patterns, instead of just using Question 2(b) and 3the pattern. This is a necessary skill before being able to use variables in algebra.

Learners have to extend shape patterns and match them to number patterns Questions 3 and 4

We thought that:

The statements and questions are linked like this:

� Match each of these statements about Task 1 correctly with the questions fromTask 1.

We have given our ideas about question 5 first. We found that finding the links showed ussome of the aspects of the Assessment Standards of Outcome 2 that Task 1 focuses on. Theyalso give a sense of the other mathematical knowledge embedded in the questions. Wefound that by looking in this sort of detail at the questions in Task 1, we were better able toanswer the more general questions that were asked about the task as a whole.


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We thought that:

� The task gives learners several opportunities to practise completing patterns,consolidate this ability and achieve what the first Assessment Standard for Grades 7and 8 requires (‘the learner investigates and extends numeric and geometric patternslooking for a relationship or rules’).

� The patterns in the task include some that are in ‘diagrammatic form’, some that do notinvolve a constant difference or ratio (question 1(i), (j), (k)) and some that are‘represented in tables’. These are some, but not all of the kinds of patterns noted underthis first Assessment Standard.

� Questions 2(b) and 3 require the learner to describe an observed relationship in words.These questions link to the second Assessment Standard. If a learner can explainpatterns using words, it gives an insight into their reasoning. Reflecting on how theclass as a whole manages Questions 2(b) and 3 will help you when you teach this aspectto them. (Assessment Task 2 deals more thoroughly with this)

If a learner performs well on this task, this may show that he or she is able toextend simple patterns. However, in this task alone, we have not gathered enoughreliable and valid evidence yet to say that the learner has achieved LearningOutcome 2 as a whole. Can you see that several of the Assessment Standards havenot been included? In your planning for the year, you will need to be sure that foreach grade you teach, all the relevant Assessment Standards have been covered –but they do not all have to be covered in one assessment task. Remember howMrs Mothae thought about this in her planning.

� What can Assessment Task 1 tell you about the learner’s ability to meet theAssessment Standards from Learning Outcome 2?

We thought that:

� The number skills of earlier grades (covered in Learning Outcome 1) are needed toinvestigate and complete the patterns set in Assessment Task 1. We thought this was agood idea as it meant that the task can help identify learners with problems with thesenumber skills.

� Some of the questions required learners to show that they understood that zero is anumber in its own right. We think it is important that learners understand this, andthat zero does not merely mean the absence of an amount. So, we thought it was a goodidea to include several questions for which the answer is zero in patterns as there is asurprising reluctance to write zero as an answer.

� What other Maths knowledge does the learner need in order to completeAssessment Task 1?


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It is important to adapt learning materials and the way you use themaccording to the needs of your classes - and this is true, too, of Task 1 and youruse of it. If you haven’t yet covered a certain concept that is in Task 1 at a timewhen you want to use it for formal assessment, then you need to change thetask accordingly. Leave out those items – replace them with something you havecovered. Or you may decide to spend class time on that concept before givinglearners the task. While you do need to be sure that all of the AssessmentStandards for a grade are covered in the year, it is pointless rushing on to coveran Assessment Standard if you have not given learners enough opportunity tounderstand and achieve those standards which learners need to build the newstandard on. Thus, if learners struggle with Assessment Standards from Outcome 1, such as those dealing with fractions, for example, they will probablynot be able to recognise patterns involving these and you will need toconsolidate this learning first.

When you are marking the learners’ work, you need to keep the different skillsand knowledge involved in mind. Although the primary focus in Task 1 is onlearners’ ability to recognise and complete patterns, at the same time it assesseslearners’ understanding of other mathematical knowledge and skills. You need tointerpret learners’ answers – to see exactly what it is they can and cannot do.Are they understanding patterning, but struggling with say, fractions? Are theyable to add time accurately, but unable to see the patterns in the time question?Your analysis of this will help you know how to help the learners progress.

� It could be used as a worksheet rather than as a formal assessment task. Using the taskin this way would give more opportunities to assist the learners when they havedifficulties. We thought this would be particularly appropriate in a Grade 7 class.

� The task could also be used as a baseline assessment for Grade 9 learners who shouldbe familiar with all the work covered. The task could provide a useful starting point forbeginning algebra in Grade 9. It would give valuable information about the Grade 9learners and their areas of difficulty.

� Some number concepts such as negative numbers and complex fractions, which arerequired in Grade 8, are not included in the task. The fraction knowledge expected hereis simple. We thought this was a good idea as the main purpose of the exercise is to testpattern extension – and we would not have wanted to handicap learners who lackedmore advanced fraction knowledge.

� Suggest different ways in which you could use this assessment task in yourclassroom.

Mrs Mothae used this task to assess her Grade 8 learners after two weeks of learning and teaching. It providedher with a formal assessment of the learners’ achievement. But it could be used in other ways, too.

We thought that:


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As we have shown, the way learners answer the questions in Task 1 will give valuableinformation about how well they have achieved the learning that preceded their doing thetask. Before such an analysis is possible, though, the work has to be marked. In order tomake sense of Mrs Mothae’s work, we would like you to complete the task below

Mark three learners’ work

Pedro, Thembeka and Ahmed’s completed assessment tasks are shown on thenext pages. Of course, there will be a greater variety of learner responses thanthose shown here in any class. However, these three represent many of the mostlikely learner responses.

1. Mark these learners’ work as if they were in your class. Refer to the markingmemorandum on page 36. To help each learner, write comments onto theirscripts that will help them to see what they need to work on. Calculate theirtotal mark out of 50.

� MARKING ASSESSMENT TASK 1

I would like you to meet three of my learners,

Ahmed, Thembeka and Pedro. They have

completed Task 1



19This work is licensed under the Creative Commons Attribution 3.0 Unported licence.

To view a copy of this licence visit http://creativecommons.org/licenses/by/3.0/

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Compare Mrs Mothae’s marking with yours

Thembeka – 33

Ahmed – 31

Pedro – 26

� Compare Mrs Mothae’s marking of the three learners’ papers with yours.Although your marks may differ slightly from hers, they should be similar.

� We have shown the comments she made on Pedro’s script. In what ways doyou think they are helpful? How could they be made more helpful?


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Focusing on individual learners

A formal assessment needs to be recorded as part of your measure of the learner’s progressthrough the year. However, recording is not enough on its own. You need to use the resultsto understand each learner’s areas of difficulty and to plan for the next cycle of teachingand learning. In marking the three learners’ work, you have allocated them an overall markfor the task as a whole. How helpful is this? The questions below will help you think aboutthis further.

� RECORDING AND INTERPRETING THE RESULTS

Thinking about using marks to record assessment

1. Does the learner’s total mark indicate to what extent she is progressingtowards achieving the outcome? In other words, does the mark show to whatextent she is successfully completing patterns?

2. Does the learner’s total mark indicate where she has managed well and whereshe has struggled?

3. Can the marks help you to identify misunderstandings or gaps in the learner’sknowledge that might have led to incorrect answers?

4. Is there a way of marking this test that would show how the learner has donein each separate type of question? What could you do to record informationabout learner achievement that would be more helpful than a mark alone?



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Using a more qualitative system of recording

1. Use Mrs Mothae’s system and your marking to complete the records forAhmed and Thembeka in Table 2 on the next page.

2. Then use the records and the learners’ scripts to help you to identify eachlearner’s strengths and weaknesses, and any gaps in knowledge ormisunderstandings.

3. Suggest what information this system gives about the learners that onlyrecording marks does not.

Mrs Mothae found that the marks gave her a general idea of whether each learnercould complete patterns, but they did not show her a learner’s areas of strength andweakness, or where the gaps in their knowledge might be.Mrs Mothae decided to identify and record how learners had done in the differentskills that were being tested in Task 1. She used these as criteria to help her assessto what extent her learners had achieved the outcomes she had set for the task. Wewill focus on choosing criteria and using them for recording assessment in moredetail in another issue of this series of booklets, but you may find it useful to thinkabout her system here.

She noted 6 criteria that focus on patterning skills, and the questions that gave herevidence of learners’ achievement of these. These are:

1. Extend simple patterns to the right 1(a) – (j)

2. Extend simple patterns to the left 1(a) – (j)

3. Extend patterns horizontally and vertically 2(a)

4. Describe simple patterns verbally 2(b) and 3(d)

5. Complete a pattern in table form 3(a)

6. Extend shape patterns 4

Mrs Mothae also chose 4 criteria to help her assess whether any learners need helpwith an understanding of fractions, decimals, large numbers or time.

7. Use fractions to complete patterns 1(c), (d), (e)

8. Use larger numbers to complete patterns 1(f) and (g)

9. Use place value and decimal numbers correctly 1(i)

10. Extend patterns of time 1(j)

Although this way of recording learner achievement looks complicated at first, it canbe efficient and is not as complicated as it seems! Mrs Mothae kept her class list nextto her and ticked off what her learners could do as she marked. Using this method,she decided not to use the marks at all! On the next page, you will see how MrsMothae completed her records for Pedro and some other learners.


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Analysing assessment information

Read what Mrs Mothae noticed about three of her learners.

1. Think about how her analysis compares with yours.

2. What would you say about Portia, Natalie and Kobus?

� Each column gives you immediate information about how a particular learnerhas done on the set of criteria. This shows each learner’s strengths andweaknesses across the range of skills and knowledge being assessed. It alsogives you a picture of whether the learner has achieved what the whole taskwas assessing – can the learner extend and describe patterns?

� Each row indicates to you whether your class has grasped a particular skill or ifthey still need more work on this.

Table 2: Mrs Mothae’s record sheet

The learner can

1. Extend simple patterns to the right 1(a) – (j)

2. Extend simple patterns to the left 1(a) – (j)

3. Extend patterns horizontally and vertically 2(a)

4. Describe simple patterns verbally 2(b) and 3(d)

5. Complete a pattern in table form 3(a)

6. Extend shape patterns Question 4

7. Use fractions to complete patterns 1(c), (d), (e)

8. Use large numbers to complete patterns 1(f) and (g)

9. Use place value and decimal numbers correctly 1(i)

10. Extend patterns of time 1(j)

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29

The South African National Curriculum Statement provides the following codes forrecording learners’ levels of achievement of outcomes.

4 = Learner's performance has exceeded the requirements.

3 = Learner's performance has satisfied the requirements.

2 = Learner's performance has partially satisfied the requirements.

1 = Learner's performance has not satisfied the requirements.

AhmedUnderstands patterning; struggles with fractions; needs to learn timenotation; didn’t notice that the pattern to the left is different from thepattern to the right in 2(b); struggled with patterning in table form andshape patterns.

Thembeka Misunderstood that the pattern must be completed to the left as well inquestion 1 (c), (d), (e); struggled with large numbers; needs to be shownthe pattern in (h); good work on shape patterns and tables (althoughneeds help on 4(b)).

Pedro Missed the pattern because he didn’t look at all the numbers in 1(a) and(b); recognised patterns in (c) to (f) but wrote down rules instead ofcompleting the pattern; needs to be reminded about conventions fordecimal numbers and for time; stronger on finding the rule, shapepatterns and completing the table.

Mrs Mothae used the information about each learner she has gathered thus far to decide on a generalassessment of how well each learner is capable of completing and describing patterns (Learning Outcome 2)at this stage in her learning programme. She used ratings of 1 to 4 to record this general assessment of howwell each learner has achieved this outcome. She also used the information she gained from the assessmentof their work to write a comment about each learner’s particular strengths and weaknesses. The table belowshows how she did this.

Completing and describing number patterns

Completing and describing shape patterns

Comments

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Ped

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Ko

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Port

ia

2 4 3233

2 3 2232

Nee

ds

pra

ctic

e u

sin

g t

able

s an

d s

hap

e p

atte

rns

Stru

gg

les

wit

h la

rge

nu

mb

ers

& c

om

ple

tin

g

pat

tern

s to

left

Can

rec

og

nis

e n

um

ber

p

atte

rns,

bu

t ca

n’t

co

mp

lete

th

em

Table 3: Mrs Mothae’s rating sheet


30


Mrs Mothae looked at her learners’ results to see if she could identify trends within theclass as a whole. These would help her to decide how to move forward.

Focusing on the group as a whole

Finding trends

1. Which questions did the three learners, Pedro, Ahmed and Thembeka, all docorrectly? What does this indicate about their knowledge and skills?

2. Which questions did these three learners do poorly? What does this indicateabout knowledge and skills that they are lacking?

It is difficult to analyse general trends from only three learners’ work. When you dothis exercise with a larger group, the trends will be clearer. However, even with onlythese learners’ work to consider, some general trends are clear.

� All three learners identified most of the number patterns in question 1, althoughonly Ahmed saw the pattern in 1(h).

� They all saw the patterns in the grid in question 2, although they confusedmultiplying and dividing when asked to describe the patterns.

� Two of the learners completed the table correctly.

� All three learners were unable to do 1(c) and (d) where they needed to movefrom whole numbers to fractions;

� Two learners struggled with large numbers in 1(g);

� Two learners struggled with writing time correctly in 1(j);

� They all struggled to complete the gaps on the left of the patterns correctly,although they were able to continue them to the right.

� All the learners struggled with the shape patterns.

Thinking about what help learners need

1. In what areas of the work do you think the learners in Mrs Mothae’s classneed help? Have a look again at the grid showing which standards each ofthe 6 learners has achieved to see the trends more clearly.

There are three areas in which learners need help � continuing patterns to the left; � continuing shape patterns � using fractions.

The learner can1. Extend simple patterns to the right 1(a) – (j)2. Extend simple patterns to

the left 1(a) – (j)3. Extend patterns horizontally and vertically 2(a)4. Describe simple patterns

verbally 2(b) and 3(d)5. Complete a pattern in table form 3(a)

6. Extend shape patterns Question 4

7. Use fractions to complete patterns 1(c), (d), (e)8. Use large numbers to

complete patterns 1(f) and (g)9. Use place value and decimal numbers correctly 1(i)10. Extend patterns of time

1(j)

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31

� USING THE ASSESSMENT RESULTS TO PLAN

An essential part of the teaching and learning cycle is deciding how to use the knowledge you havegained from assessing the learners (Step 4 on pages 9 and 10) You need to do two things.

� Firstly, think about what the assessment suggests about how you might better teach thissection of work another time. What strengths and weakness can you now anticipate, andhow can you use this knowledge to plan your teaching better?

� Secondly, think about what you can do to help the learners whose work you have justassessed. How should you adjust your teaching plan to support their learning better?You cannot spend too long on going over work that has not been understood, but it canbe damaging to the learners’ progress to just ignore their difficulties. They are likelyto struggle with the next section of the work. How can you help both individuals andthe group as a whole? How can you extend those learners who are clearly able to do thework well?

Think about what to do to support learning

1. Make suggestions about how you would address the problems that havebeen identified in the learners’ work.

2. Compare your ideas with what Mrs Mothae did.

This is what Mrs Mothae did to address these areas of concern:

� She showed the class how the patterns that they completed in the task continuein both directions. She helped the class see that the rule they find if they work tothe right is the inverse of the rule they will use if they work to the left. She gavethem some additional examples to work on in pairs to check that they hadunderstood.

� She realised that she could not just give learners different versions of the sameshape patterns, so she spent a whole lesson letting learners explore shapepatterns by building the patterns with cardboard shapes, investigating theiranswers and explaining them in groups. This gave them a concrete experience ofshape patterns.

� She had to remind them of previous work done with fractions and gave them ahomework exercise on adding and subtracting simple fractions.

� She offered to make an extra time outside of classroom time to work with twolearners who still struggled. She built up their confidence by working withsimpler patterns first.

� She set some tasks to stretch the capabilities of the strongest learners, andarranged for them to work together in a group on some occasions when othergroups of learners worked on simpler tasks.


32


After some work such as that described above, Mrs Mothae was aware that she needed tomove on to the next step in her plan – verbal and algebraic expression of patterns (iegeneralisations). She continued with the planned classwork, but included more examplesof shape patterns and patterns involving fractions. The learners worked in pairs and hadto keep explaining what they were doing while they worked. They also began using lettersymbols for variables to generalise the patterns. Mrs Mothae assessed them informally,mediating, assisting and observing their progress.

By the time she used Assessment Task 2, most of her learners were able to explain and usenumber and shape patterns, although some still struggled with using variables. Those whostill struggled with fractions attended extra lessons after school.

In Assessment Task 2, Mrs Mothae set questions that asked learners to explain therelationship between different numbers in patterns. If you look at this task in Appendix 2,you will see that some of the same patterns of Assessment Task 1 are asked in a similarway in order to consolidate their knowledge. From question 2 on, the learners may startby using words to explain, but by the end should be able to show that they can usemathematical operations and variables (this addresses steps 2 and 3 identified on page 6).

As before, if the learners’ performance on Assessment Task 2 showed that they could dothis work, then Mrs Mothae felt that she could move on to the work planned next, leadingup to Assessment Task 3 and finally Assessment Task 4. Learners need to have workedwith flow diagrams and graphs before they can complete Assessment Task 3, and MrsMothae made sure she gave them plenty of opportunity to do this, mediating their learning.

Assessment Task 4 assesses learners’ ability to apply their understanding to a problem, andagain, learning opportunities were provided for this. While learners worked, Mrs Mothaechecked on how they were doing, made informal notes about their progress and intervenedto help and extend them where appropriate. As we described for Task 1, after each of thefollowing assessment tasks Mrs Mothae checked more formally on what learners’ worktold about their achievement of the Assessment Standards their work was focusing on, andtheir strengths and weaknesses, and spent some time dealing with aspects of concernbefore moving on. She also built additional support of work already covered into herplanned new work to help consolidate learning where she now realised this was needed.

You may want to use these assessment tasks in your teaching and learning cycle, adaptingthem to suit your needs

� THE CYCLE IS ON-GOING

We hope that you found this booklet useful. It showed you how one teacher tried to useassessment as part of the teaching and learning cycle. Mrs Mothae did not find it easy, butshe enjoyed trying new ideas - and then changing them if she finds something that seems towork better. We hope that you will be encouraged to do the same.

Conclusion



33

34

38

42

46

Appendices 1 - 4: The Assessment Tasks and Answers

Task 1Focuses on learners’ ability to recognise and complete a

variety of types of patterns in number and shape

Task 2Relates to learners’ ability to express patterns in words

and to complete a variety of patterns that are

represented in tables

Task 3Assesses learners’ ability to express relationships and

generalisation using flow diagrams and graphs

Task 4Assesses learners’ ability to apply their knowledge of

patterning, generalisation and algebra to solve an

integrated, ‘real life’ problem.


34


Assessment Task 1

i) from left to right: ………………………………. ii) from right to left: ………………………………......

iii) from top to bottom: …………………………. iv) from bottom to top: …………………………….....

Note: Marks will be awarded for showing that you can complete or describe the patterns correctly, and for the accuracy of your answers.

Total Marks: 50

1. Fill in the missing numbers to complete these patterns (24 marks)

a) 15; 30; 45; ____; _____; _____.

b) 72; 65; 58; ____; _____; _____.

c) _____; 1; 4; 16; 64; _____.

d) _____; 81; 27; 9; 3; _____; _____.

e) _____; ; ; ; _____; _____.

f) _____; 789; 678; 567; ______.

g) ______ 21 079; 21 179; _______; 21 379

h) ______; 1; 2; 4; 7; _____; _____.

i) ______; 2,05; 2,1; 2,15; ______; ______.

j) 10h50; 11h40; 12h30; _______; ______.

2a) The numbers in this grid are arranged in patterns. Use the patterns to find the values of A, B, C, D and E (1 x 5 = 5 marks)

162 A

54 108 CB

E 4D

27

123

189

A =

B =

C =

D =

E =

b) Describe the patterns in 2(a): (1 x 4 = 4 marks)

41

21

43



35

3a. Gogo Zuma loves to cook her food in a three-legged pot over the fire. She discovers that many of her neighbours also use these pots. Work out how many legs there are on the number of pots listed in the table below. (3 marks)

3b) Explain how you would work out what the number of legs is for any number of pots. (2 marks)

4. For each pattern below: a) draw the part that comes immediately before and immediately after the three parts shown here. (3 x 2 = 6 marks) b) complete the matching number pattern for each shape. (3 x 2 = 6 marks)

Number ofPots 1 2 3 4 8 13 50 200

Number ofLegs

........................................................................................................

Pattern A

4 9 16

Pattern B

8 12 16

Pattern C

8 10 12


36


Assessment Task 1 with answers and some of the working in bold

i) from left to right: Multiply by 2 ii) from right to left: Divide by 2

iii) from top to bottom: Divide by 3 iv) from bottom to top: Multiply by 3

Note: Marks will be awarded for showing that you can complete or describe the patterns correctly, and for the accuracy of your answers.

Total Marks: 50

1. Fill in the missing numbers to complete these patterns (24 marks)

a) 15; 30; 45; 60; 75; 90

b) 72; 65; 58; 51; 44; 37

c) ; 1; 4; 16; 64; 256 d) 243; 81; 27; 9; 3; 1; e) 0; ; ; ; 1; 1

f) 900; 789; 678; 567; 456

g) 20 979; 21 079; 21 179; 21 279; 21 379 h) 1; 1; 2; 4; 7; 11; 16

i) 2; 2,05; 2,1; 2,15; 2,2; 2,25

j) 10h50; 11h40; 12h30; 13h20; 14h10

2a) The numbers in this grid are arranged in patterns. Use the patterns to find the values of A, B, C, D and E (1 x 5 = 5 marks)

162 A

54 108 CB

E 4D

27

123

189

A = 324

B = 13 or 13,5

C = 216

D = or 0,5

E = 2

b) Describe the patterns in 2(a): (1 x 4 = 4 marks)

1 mark for pattern, 1 for correct answer throughout. Extra mark for such things as pattern to left and for pattern with 0 where indicated.

(pattern: +15) 2 marks

(pattern: -7) 2 marks

(pattern: ÷ 3) 3 marks ( in last space)

(pattern: x 4) 3 marks ( in first space)

(pattern: + ) 3 marks (0 in first space)

(pattern: -111) 2 marks

(pattern: +100) 2 marks

(pattern: +0, +1, +2...) 3 marks (1 in first space)

(pattern: +0.05) 2 marks

(pattern: +50mins) 2 marks (check format)

14

14

14

14

34

12

12

12

13

1413

40,5 81 648

4,5 72

1,5 6 24

81

36



37

3a. Gogo Zuma loves to cook her food in a three-legged pot over the fire. She discovers that many of her neighbours also use these pots. Work out how many legs there are on the number of pots listed in the table below. (3 marks)

3b) Explain how you would work out what the number of legs is for any number of pots. (2 marks)

4. For each pattern below: a) draw the part that comes immediately before and immediately after the three parts shown here. (3 x 2 = 6 marks) b) complete the matching number pattern for each shape. (3 x 2 = 6 marks)

Number ofPots 1 2 3 4 8 13 50 200

Number ofLegs

Multiply the number of pots by 3

Pattern A

4 9 16

Pattern B

8 12 16

Pattern C

8 10 12

3

1

4 20

6 14

25

6 9 12 24 39 150 600


38


Assessment Task 2

1. Complete the patterns below

a) ____; ____; 1; 4; 16; _______.

b) _____; ______; _______; 3; 9; 27.

c) 1; ____; 2; 4; 7; _____; _____; 22.

d) 26; 18; 12; ______; ______; 6.

e) ____; 1 ; 2 ; 3 ; _____; _____.

f) ____; _____; 10h45; 11h30 12h15; ______; _____.

g) _____; 4,055; 4,155; 4,255; ______.

h) 2 250 m; 2 km; _____; _____; 1250 m; 1 km.

i) _____; ______; 1; 3; 5; 7; _____.

2a) Explain how you would find:

3) Three glasses can be poured from one litre of milk. How many glasses of milk can be poured from a given number of litres?

4) One car needs five tyres including the spare. How many cars can be fitted with a given number of tyres?

a) the number of sides on a given number of squares..........................................................................

5) Four people can sit around a square table like this one.:

6) If two of the same tables are pushed together, then 6 people can sit around them

a) How many people can sit around two tables set apart

b) How many people can sit around 5 tables set apart

c) Describe in words how you would find the number of people that could sit at any number of tables set apart.

b) the area of a square of any given side length...................................................................................

15

25

35

............................................................................

............................................................................



39

7. Look at the pattern in the shapes and in the numbers of degrees on each shape.

8. In this pattern, three matches form 1 triangle, five matches form 2 triangles.

a) Draw Shape 3 in the space aboveb) Find the number of degrees shown by the arrow in Shape 3c) How many degrees would the arrow show if there were a Shape 6d) How could you find the angle shown by the arrow in any shape in the same pattern?

a) Draw and complete a table to show the number of matches needed to form 1, 2, 3, 4, 20 and 50 triangles

b) Describe how you would find the number of matches needed to form any number of triangles in this way.

c) Draw a line graph to show the relationship between the number of triangles (horizontal axis) and the number of matches (vertical axis). You can use the grid given here.

a) Complete the mathematical table below to show how many people can sit around the given number of tables pushed together.

b) Describe how you would find the number of people that could sit at any number of tables pushed together.

Number ofTables

Shape 1 Shape 2 Shape 3 Shape 4120 90 60

1 2 3 4 8 15

Number ofPeople

........................................................................................................

..........................................................................................

120 90 5160

Shape 551

0Number of Triangles

Num

ber

of M

atch

es

123456789

10

1 2 3 4 5


40



1. Complete the patterns below

a) ; ; 1; 4; 16; 64.

b) ; ; 1; 3; 9; 27.

c) 1; 1; 2; 4; 7; 11; 16; 22.

d) 26; 18; 12; 8; 6; 6.

e) 0; 1 ; 2 ; 3 ; 4 .

f) 09h15; 10h00; 10h45; 11h30; 12h15; 13h00; 13h45.

g) 3,955; 4,055; 4,155; 4,255; 4,355.

h) 2 250 m; 2 km; 1750m; 1500m (or 1,5km); 1250 m; 1 km.

i) -3; -1; 1; 3; 5; 7; 9.

2a) Explain how you would find:

3) Three glasses can be poured from one litre of milk. How many glasses of milk can be poured from a given number of litres?

4) One car needs five tyres including the spare. How many cars can be fitted with a given number of tyres?

a) the number of sides on a given number of squares 4 x number of squares

5) Four people can sit around a square table like this one.:

6) If two of the same tables are pushed together, then 6 people can sit around them

a) How many people can sit around two tables set apart?

b) How many people can sit around 5 tables set apart?

c) Describe in words how you would find the number of people that could sit at any number of tables set apart.

b) the area of a square of any given side length Length x length (or )

25

15

35

3 x numbers of litres

2 x 4 = 8 people

5 x 4 = 20 people

Number of people is 4 timesthe number of tables

Number of tyres divided by 5

116

45

19

13

14

2



41

7. Look at the pattern in the shapes and in the numbers of degrees on each shape.

8. In this pattern, three matches form 1 triangle, five matches form 2 triangles.

a) Draw Shape 3 in the space aboveb) Find the number of degrees shown by the arrow in Shape 3 c) How many degrees would the arrow show if there were a Shape 6? d) How could you find the angle shown by the arrow in any shape in the same pattern?

a) Draw and complete a table to show the number of matches needed to form 1, 2, 3, 4, 20 and 50 triangles

c) Draw a line graph to show the relationship between the number of triangles (horizontal axis) and the number of matches (vertical axis). You can use the grid given here.

a) Complete the mathematical table below to show how many people can sit around the given number of tables pushed together.

b) Describe how you would find the number of people that could sit at any number of tables pushed together.

Number ofTables

Shape 1 Shape 2 Shape 3 Shape 4120 90 60

1 2 3 4 8 15

4 6 8 10 18 32Number ofPeople

Learners answers will vary. One correct answer is: 'twice the number of tables and add 2'

120 90 516072

Shape 551

0Number of Triangles

Num

ber

of M

atch

es

123456789

10

1 2 3 4 5

Angle will be + 2 or

Number of triangles 1 2 3 4 20 50Number of matches

(Number of triangles x 2)+1 or 1+(2 x number of triangles)

3 5 7 9 41 101

b) Describe how you would find the number of matches needed to form any number of triangles in this way.

360 = 458

360 number of sides

360 shape number


42


Assessment Task 3

1. Work out:

a) How many days in weeks?

b) How many hours do you sleep in m days (If you sleep 8 hours each day)

c) How many internal angles are there in t squares?

d) What is the surface area of a cube with edge length ?

e) How many legs on: i) p chickens?

ii) q cows?

iii) r snakes?

f) A taxi holds people. How many people can fit into taxis?

g) How many cuts do you make to divide a sausage into pieces?

h) How many quarters in k?

a) Put a circle around all the equations below that show Flow Diagram I

b) In Flow Diagram I: i) if = 3, what is ? ii) If = -3, what is ?

2. Look at the two flow diagrams below.

3. a) Draw a flow diagram to show = (2 + 3)

b) If = 9, what is ? c) If = 9, what is ?

I.

II.

÷3 +5

+5 ÷3

=3 + 5 ( + 5)=

3= 3

+ 5

c) In Flow Diagram II: i) if = 3, what is ? ii) If = -3, what is ?

= 5 ÷ 3= + 5 ÷ 3 = ÷ 3 + 5 = ÷ (3 + 5)

2

= ( )+ 5 ÷ 3



43

4

a) How many small cubes are needed to make a cube of edge length 3?

b) Complete the table below to show the number of little cubes needed to build a large cube.

To make a cube of edge length 1, you need one small cubeTo make a cube of edge length 2, you need 8 small cubes

Edge length 1

1

2 3 4 5

Number of small cubes needed

d) On squared paper, draw a line graph

to show the relationship between the

edge length of a large cube

(horizontal axis) and the number of

little cubes you need to form a

the large cube (vertical axis).

You can use the grid given here.

c) How would you find the number of small cubes needed to create a larger cube of any edge length?

LENGTH OF EDGE OF LARGE CUBE (counted in small cubes)

NU

MB

ER O

F SM

ALL

CU

BES

(n

eed

ed t

o m

ake

up

a la

rge

cub

e)


44



3. a) Draw a flow diagram to show = (2 + 3)

b) If = 9, what is ? c) If = 9, what is ?

or

1. Work out:

a) How many days in weeks? 7 days

b) How many hours do you sleep in m days (If you sleep 8 hours each day) 8m hours

c) How many internal angles are there in t squares? 4t internal angles

d) What is the surface area of a cube with edge lenghth ? Surface area is 6

e) How many legs on: i) p chickens? 2p legs

ii) q cows? 4q legs

iii) r snakes? 0 (snakes have no legs!)

f) A taxi holds people. How many people can fit into taxis? people

g) How many cuts do you make to divide a sausage into pieces? -1 cuts

h) How many quarters in k?

a) Put a circle around all the equations below that show Flow Diagram I

2. Look at the two flow diagrams below.

I.

II.

÷3 +5

+5 ÷3

b) In Flow Diagram I: i) if = 3, what is ? ii) If = -3, what is ?

=3

3+ 5 = 1 + 5 = 6 =

3

-3+ 5 = -1 + 5 = 4

2

2

=3 + 5 ( + 5)=

3= 3

+ 5

= 5 ÷ 3= + 5 ÷ 3

= ( )+ 5 ÷ 3

= ÷ 3 + 5 = ÷ (3 + 5)

2 = -6 2 +3 = -3

= -3

4k

= = 2=3 3

238

c) In Flow Diagram II: i) if = 3, what is ? ii) If = -3, what is ?(3 + 5) =

3 32 = (-3 + 5)

= [(2 x 9)] +3]

= (21)

= 441

2

2

9 = (2 +3)

3 =

2 = 0

2 +3

= 0

2

+-2 +3 = 3

9 = (2 +3)2

x 2 + 3 all squared



45

4

a) How many small cubes are needed to make a cube of edge length 3?

b) Complete the table below to show the number of little cubes needed to build a large cube.

c) How would you find the number of small cubes needed to create a large cube

of any edge length?

d) On squared paper, draw a line graph to

show the relationshop between the

edge length of a large cube (horizontal axis)

and the number of little cubes you need to

form a the large cube (vertical axis).

To make a cube of edge length 1, you need one small cubeTo make a cube of edge length 2, you need 8 small cubes

Edge length 1

1 8 27 64 125

2 3 4 5

Number of small cubes needed

27 small cubes

3 3(Edge length) =

LENGTH OF EDGE OF LARGE CUBE (counted in small cubes)

NU

MB

ER O

F SM

ALL

CU

BES

(n

eed

ed t

o m

ake

up

a la

rge

cub

e)


46


Assessment Task 4

1. Here is a plastic cup.

Its total height is 11cm The rim of the cup is cm high.

2. Here are two rough sketches of rectangles that have a perimeter of 36cm.

a) Draw another 10 different rectangles (including a square), all of which have a perimeter of 36cm. Use 17cm as the longest possible length of a side and 1cm as the shortest possible length of one side

12

12 cm rim

11cm

a)

When 3 cups are stacked inside each

other, they look like this.

i) What is the total height of 7 cups stacked in the same way?

ii) The total height of some cups stacked in this way is 17cm. How many cups are there in the stack?

iii) Write an algebraic formula to calculate the height of cups stacked in this way.

b) The same cups are stacked in the pattern shown here:

i) Write an algebraic formula to find the height of cups stacked in this way

ii) Find the difference in height between cups stacked as shown in (a) and cups stacked as shown in (b)

Perimeter:distance around theedge ofa shape 17cm

1cm 3cm

15cm



47

iii) ; ; 2 ;

3. This famous number pattern below is called the Fibonacci sequence. To find the next number you add the previous two.

b) Find the area of each of your rectangles.

c) Copy and complete the table below to show the relationship between the length of one of the sides of each rectangle and its area.

17

17

15

45Area

e) Write a sentence to describe how the area of the rectangle changes as changes.

a) Find the next two numbers in the series.

c) Use the same rule as the one for the Fibonacci series to find the missing numbers in the following serries.

d) Plot the points from your table on a graph to show the relationship between the length of in centimetres, and the area in cm . You can join these points because can have any real value (

2

R).

1 1 2 3 5 8 13

b) Here are more Fibonacci numbers. Find the missing two.

89 144 233 377

i) 2,17; 4,05; 6,22;

v) 5; -2; 3;

ii) 155 000; 2 million;

iv) 4 ; 6 ;

21

+ +2 2


48



11 - (10 + )

= 10 + or 11 + ( )cm

1. Here is a plastic cup.

Its total height is 11cm The rim of the cup is cm high.

2. Here are two rough sketches of rectangles that have a perimeter of 36cm.

a) Draw another 10 different rectangles (including a square), all of which have a perimeter of 36cm. Use 17cm as the longest possible length of a side and 1cm as the shortest possible length of one side.

12

12 cm rim

11cm

a)

When 3 cups are stacked inside each

other, they look like this.

i) What is the total height of 7 cups stacked in the same way?

ii) The total height of some cups stacked in this way is 17cm. How many cups are there in the stack?

iii) Write an algebraic formula to calculate the height of cups stacked in this way.

b) The same cups are stacked in the pattern shown here:

i) Write an algebraic formula to find the height of cups stacked in this way

ii) Find the difference in height between cups stacked as shown in (a) and cups stacked as shown in (b)

Perimeter:distance around theedge ofa shape 17cm

1cm 3cm

15cm

10 + (7 x ) 10 + 3 = 14cm or 11 + (6 x ) 11 + 3 =14cm 12

12

12

12

12

- 12

12

12

12

12

13 cups

11

There are 17 possible rectangles altogether. We have not drawn them here, but their dimensions are shown in the table. Learners will have chosen 10 from this set, but we have shown them all so as to include whichever any learner chooses. They do not have to show both sides – but we have done so to provide the range of answers possible in 2c. All learners should include the rectangle of sides 9 x 9 cm, as this will give the square asked for in the question.



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As increases from 1 to 9, area increases. As increases for > 9, the area decreases.

iii) ; ; ; 2 ;

3. This famous number pattern below is called the Fibonacci sequence. To find the next number you add the previous two.

b) Find the area of each of your rectangles.

c) Copy and complete the table below to show the relationship between the length of one of the sides of each rectangle and its area.

17

17

15

45Area

e) Write a sentence to describe how the area of the rectangle changes as changes.

a) Find the next two numbers in the series.

c) Use the same rule as the one for the Fibonacci series to find the missing numbers in the following serries.

d) Plot the points from your table on a graph to show the relationship between the length of in centimetres, and the area in cm . You can join these points because can have any real value (

2

R).

1 1 2 3 5 8 13

b) Here are more Fibonacci numbers. Find the missing two.

89 144 233 377

i) 2,17; 4,05; 6,22;

v) -7; 5; -2; 3; 1

ii) 155 000; 2 million;

iv) ; 4 ; 6 ;

21

+ + +22 2

1 2 3 4 5 6 7 8 9 10 11 12 13 15 1614

16 14 13 12 11 10 9 8 7 6 5 3 24

32 56 65 72 77 80 81 80 77 72 65 45 32

17

1

1756

21 34

55

1,88;

2 345 000; 2 655 000

10,27

610

3

22

10

2


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Notes:
